Abstract

We generalize r-costar module to r-costar pair of contravariant functors between abelian categories.

1. Introduction

For a ring , a fixed right -module , and , let fgd-tl() denote the class of torsionless right -modules whose -dual are finitely generated over and fg-tl () denote the class of finitely generated torsionless left -modules. is called costar module if
is a duality. Costar modules were introduced by Colby and Fuller in [1]. is said to be an r-costar module provided that any exact sequence
such that and are -reflexive, remains exact after applying the functor if and only if is -reflexive. The notion of r-costar module was introduced by Liu and Zhang in [2]. We say that a right -module is -finitely -copresented if there exists a long exact sequence
such that are positive integers for . The class of all -finitely -copresented modules is denoted by -. We say that a right -module is a finitistic -self-cotilting module provided that any exact sequence
such that - and is a positive integer, remains exact after applying the functor and --. Finitistic -self-cotilting modules were introduced by Breaz in [3].

In [4] Castaño-Iglesias generalizes the notion of costar module to Grothendieck categories. Pop in [5] generalizes the notion of finitistic -self-cotilting module to finitistic --cotilting object in abelian categories and he describes a family of dualities between abelian categories. Breaz and Pop in [6] generalize a duality exhibited in [3, Theorem 2.8] to abelian categories.

In this work we continue this kind of study and generalizes the notion of r-costar module to r-costar pair of contravariant functors between abelian categories, by generalizing the work in [2]. We use the same technique of proofs of that paper.

2. Preliminaries

Let and be additive and contravariant left exact functors between two abelian categories and . It is said that they are adjoint on the right if there are natural isomorphisms
for and . Then they induce two natural transformations and defined by and . Moreover the following identities are satisfied for each and :
The pair is called a duality if there are functorial isomorphisms and . An object of () is called -reflexive (resp., -reflexive) in case (resp., ) is an isomorphism. By we will denote the full subcategory of all -reflexive objects. As well by we will denote the full subcategory of all -reflexive objects. It is clear that the functors and induce a duality between the categories and .

We say that the pair of left exact contravariant functors is r-costar provided that any exact sequence
with remains exact after applying the functor if and only if .

An object is called -finitely generated if there is an epimorphism , for some positive integer . We denote by the subcategory of all -finitely generated objects. denotes the class of all summands of finite direct sums of copies of . We will denote by the full subcategory of all projective objects in .

From now on we suppose that has enough projectives that is, for every object there is a projective object and an epimorphism . It is clear that we can construct a projective resolution for any object . Suppose we have a projective resolution of
This gives rise to the sequence
and the cochain complex , which we can compute its cohomology at the th spot (the kernel of the map from modulo the image of the map to ) and denote it by . We define as the th right derived functor of . For the functor we define for every .

Let
be an exact sequence in . Applying the functor we get the exact sequence
where . Let be the canonical decomposition of , where is the inclusion map. Applying the functor to the sequence (2.7), we have the following exact sequence
Now if we put , then
So we have the following commutative diagram
(2.10)

3. r-Costar Pair of Contravariant Functors

We will fix all the notations and terminologies used in previous section.

Proposition 3.1. Let be a pair of left exact contravariant functors which are adjoint on the right. Assume that and . Then is an r-costar.

Proof. Let
be an exact sequence with . Assume that we have the exact sequence
after applying the functor . Applying the functor to the last sequence, we get an exact sequence
since . Hence we have the following commutative diagram:
(3.4)
Since , and are isomorphisms. Now it is clear that is an isomorphism which means that .Conversely, suppose that . Applying the functor to the sequence (3.1), we get an exact sequence
where . Hence we can get the exact sequence
for some , and is the inclusion map. Applying the functor to the sequence (3.5), we have the following exact commutative diagram (see diagram (2.10))
(3.7)
where . Note that and are isomorphisms, since . lt is clear from the diagram that . Now for all , by dimension shifting, since . Hence . Now applying the functor to sequence (3.6), we get the long exact sequence
Above we conclude that and by assumptions , thus and . Hence by dimension shifting . Now consider the following part from sequence (3.8)
Note that in diagram (3.7) is an isomorphism, since and are isomorphisms. Hence is an isomorphism, since is an isomorphism, so from sequence (3.9), We conclude that . Since by assumptions, and hence from sequence (3.6) canonically. Therefore the functor preserves the exactness of the exact sequence
in . We conclude that the pair is an r-costar.

Corollary 3.2. Let be a pair of left exact contravariant functors which are adjoint on the right. Assume that . Then is an r-costar.

Proof. Let , then . Hence .

Proposition 3.3. Let be a -reflexive generator in and . Let be an r-costar pair. If , then for any , there is an infinite exact sequence
which remains exact after applying the functor , where for each .

Proof. Let ). Then , so by assumption there is an exact sequence
Applying the functor we have an exact sequence
for some . Since is an r-costar pair, the last sequence is exact after applying the functor , that is we have an exact sequence
Applying the functor again we get the following commutative diagram with exact rows
(3.15)
Since , , . By repeating the process to , and so on, we finally obtain the desired exact sequence.

Proposition 3.4. Let be a -reflexive and a projective generator in and . Let be an r-costar pair and suppose that . Then .

Proof. Let , then and hence by Proposition 3.3, there is an infinite exact sequence
which remains exact after applying the functor , where for each . So we have an exact sequence
Again the last sequence remains exact after applying the functor , since we get a sequence isomorphic to sequence (3.16), because , , for each , are -reflexive. We obtain that by dimension shifting.

Suppose we have the following exact sequence in
where , are projective objects in and . Applying the functor we get the following exact sequence

Applying the functor we get the following commutative diagram with exact rows
(3.20)
If , then it is clear that ).

Proposition 3.5. Let be a -reflexive generator in . Let be an r-costar pair and suppose that . Then .

Proof. For any , we can build the following exact sequence in
where , are projective objects and an object in . By the argument before the proposition it is clear that and hence . Applying the functor we get the following exact sequence
Applying the functor we get the following commutative diagram with exact rows
(3.23)
Thus it is clear that .

Now we are able to give the following characterization of r-costar pair.

Theorem 3.6. Let be a pair of left exact contravariant functor which are adjoint on the right. Suppose that be a -reflexive projective generator in and . Then is an r-costar if and only if and .

Corollary 3.7. Let be a pair of left exact contravariant functor which are adjoint on the right. Suppose that be a -reflexive projective generator in and . If , then the following are equivalent.(1) is an r-costar.(2)For any exact sequence
with , then if and only if the exact sequence remains exact after applying the functor .

Proof. (1)(2) follows from the definition of r-costar pair.(2)(1) the proof goes the same as the proofs of Propositions 3.3, 3.4, 3.5, and Theorem 3.6.